Drorbn - User contributions [en]

Notes for wClips-120201/0:52:41

2012-02-08T16:18:44Z

Lzhang:

Comment: The target space has only quadratic relations here because we are talking about Quadratic UFTI.

Notes for wClips-120201/0:52:41

2012-02-08T16:12:09Z

Lzhang:

Q: Can someone motivate this definition of UFTI? E.g. Why use the target space which has only quadratic relations?

Notes for wClips-120201/0:52:41

2012-02-08T16:07:32Z

Lzhang:

Q: Can someone motivate this definition of UFTI?

Notes for wClips-120201/0:14:50

2012-02-08T15:39:39Z

Lzhang:

Question/Claim (not stated in the video): The set of \bar{\simga_{ij}} generate I.

Notes for wClips-120201/0:14:50

2012-02-08T15:33:01Z

Lzhang:

Question/Claim (not stated in the video): \bar{\simga_{ij}} and \bar{\sigma_{ij}^{-1}} generate I.

Notes for wClips-120201/0:14:50

2012-02-08T15:13:48Z

Lzhang:

Notes for wClips-120201/0:17:20

2012-02-08T15:12:01Z

Lzhang:

Clarification: The questions on the board refer to "What are the generators and relations of gr(QG)", where G=PvBn here.

Notes for wClips-120201/0:17:20

2012-02-08T15:11:25Z

Lzhang:

Clarification: The questions on the board refer to "What are the generators and relations of gr(QG)" where G=PvBn here?

Notes for wClips-120201/0:12:05

2012-02-08T15:10:41Z

Lzhang:

Q: What does generation mean? Where is the constraint \sum_i{q_i}=0 captured in the generation equation? -- Lucy

Notes for wClips-120201/0:14:50

2012-02-08T15:10:05Z

Lzhang:

Q: How is the semi-virtual crossing \sigma_{ij}^{-1} - 1 depicted in pictures? -- Lucy

Notes for wClips-120201/0:17:20

2012-02-08T15:03:44Z

Lzhang:

Clarification: The questions on the board refer to "What are the generators and relations of gr(QG)"?

Notes for wClips-120201/0:12:05

2012-02-08T14:55:28Z

Lzhang:

What does generation mean? Where is the constraint \sum_i{q_i}=0 captured in the generation equation?

Notes for wClips-120125-1/0:16:26

2012-02-01T16:24:20Z

Lzhang:

It seems like (in the proof below): it has not been proven that the LHS contains the RHS.
Addition: There should also be a version of Equation (2) in McCool's Theorem as in Artin's Theorem.

Notes for wClips-120125-1/0:08:11

2012-02-01T16:15:55Z

Lzhang:

Correction (See discussion from around 0:43:40): Product in equation (2) of this blackboard shot should be from n to 1.

Notes for wClips-120125-1/0:16:26

2012-02-01T16:06:40Z

Lzhang:

It seems like: it has not been proven that the LHS contains the RHS.

Notes for wClips-120125-1/0:24:44

2012-02-01T15:59:03Z

Lzhang:

Time for a little exercise.

Notes for wClips-120125-1/0:27:13

2012-02-01T14:43:25Z

Lzhang:

Correction: one too many strand on the right hand side.

Notes for AKT-090910-2/0:11:41

2009-11-11T20:04:33Z

Lzhang: video annotation

Invariance under R1: get <math>-A^3=1</math>.

Notes for AKT-090910-2/0:02:16

2009-11-11T20:03:03Z

Lzhang:

Invariance under R2 forces us to set (after computation and comparison):

: <math>B=A^{-1}</math>
: <math>d=-A^2-A^{-2}</math>

Now, the Kauffman bracket becomes a Laurent polynomial in a single variable, <math>A</math>.

Notes for AKT-090910-2/0:08:28

2009-11-11T20:00:41Z

Lzhang: video annotation

Invariance under R3: automatically satisfied, as long as the above relations between the 3 variables are satisfied (i.e. invariant under R2).

Notes for AKT-090910-2/0:01:47

2009-11-11T19:56:49Z

Lzhang:

Is the Kauffman bracket a knot invariant? Need to check invariance under the 3 Reidemeister moves (assuming invariance under planar isotopy). It turns out that R2 yields the most interesting constraints on the variables <math>A</math>, <math>B</math> and <math>d</math>. So, we will proceed in the order R2, R3 then R1.

Notes for AKT-090910-2/0:02:16

2009-11-11T19:37:12Z

Lzhang: video annotation

Invariance under R2 forces us to set (after computation and comparison):
 
<math>B=A^{-1}</math>
 
<math>d=-A^2-A^{-2}</math>

Now, the Kauffman bracket becomes a Laurent polynomial in a single variable, <math>A</math>.

Notes for AKT-090910-2/0:01:47

2009-11-11T19:34:53Z

Lzhang:

Notes for AKT-090910-1/0:51:06

2009-11-11T19:33:23Z

Lzhang:

Preview of the 2nd hour: Will derive relations between <math>A</math>, <math>B</math> and <math>d</math>, in order for the bracket polynomial to be invariant under the Reidemeister moves.

Notes for AKT-090910-2/0:01:47

2009-11-11T19:23:27Z

Lzhang: video annotation

Is the Kauffman bracket a knot invariant? Need to check invariance under the 3 Reidemeister moves (assuming invariance under planar isotopy). It turns out that R2 yields the most interesting constraints on the variables A, B and d. So, we will proceed in the order R2, R1 then R3.

Notes for AKT-090910-2/0:00:08

2009-11-11T19:18:19Z

Lzhang: video annotation

Well-defineness of the Kauffman bracket: The resulting polynomial (or the resulting zoo of trivial links) is independent of the order of smoothing.

Notes for AKT-090910-1/0:46:24

2009-11-11T19:02:25Z

Lzhang: video annotation

Cont'd: Defining the last axiom (about unlinks) of the Kauffman bracket.

Notes for AKT-090910-1/0:51:06

2009-11-11T19:00:28Z

Lzhang: video annotation

Preview of the 2nd hour: Will derive relations between A, B and d, in order for the bracket polynomial to be invariance under the Reidemeister moves.

Notes for AKT-090910-1/0:47:35

2009-11-11T18:54:06Z

Lzhang: video annotation

Example: Computing the Kauffman bracket of the Hopf link.

Notes for AKT-090910-1/0:40:07

2009-11-11T18:53:05Z

Lzhang: video annotation

Defining the Kauffman bracket or simply the bracket as a polynomial in <math>A</math>, <math>B</math>, and <math>d</math>. It is recursively defined on links.

Notes for AKT-090910-1/0:39:37

2009-11-11T18:49:53Z

Lzhang: video annotation

Historical remarks about the Jones polynomial (discovered in the 80s) and the related Kauffman bracket (later found to simplify the description of the Jones polynomial).

Notes for AKT-090910-1/0:37:17

2009-11-11T18:45:42Z

Lzhang:

Obvious limitation of Tricolourability: It only takes two possible values, so definitely cannot distinguish between three knots.
Hence, we desire a stronger invariant which can get us closer to the goal of distinguishing the many different knots listed in the Rolfsen table.

Notes for AKT-090910-1/0:37:00

2009-11-11T18:42:58Z

Lzhang:

2nd math item: Towards introducing a much stronger knot invariant, the Jones polynomial.

Notes for AKT-090910-1/0:37:17

2009-11-11T18:39:22Z

Lzhang: video annotation

Obvious limitation of Tricolourability: It only takes two possible values, so definitely cannot distinguish between three knots.

Notes for AKT-090910-1/0:37:00

2009-11-11T18:33:25Z

Lzhang: video annotation

2nd math item: Towards introducing a stronger knot invariant, the Jones polynomial.

AKT-09/To Do

2009-09-22T22:06:10Z

Lzhang: respond to a to-do item posted by myself

* Create an information page on how to annotate the lectures (creating a new page might be relatively easy, but it's not clear how to link from the associated dbnvp page when one wants to comment about a different point in the lecture). Perhaps Dror thinks it's his job?

** Answer: To create a new page for annotation, click on "Comment on [time]" just below the video screen on the video webpage.

* Add link back to AKT homepage in AKT sub-pages OR show the navigation menu in these sub-pages.

** Response: Checking back, it appears that this was already DONE for all sub-pages not intended to be closed after viewing.

Notes for AKT-090910-1/0:36:19

2009-09-22T21:36:50Z

Lzhang: video annotation

End of 1st math item: Q.E.D. Trefoil knot <math>\ne</math> Unknot.

Notes for AKT-090910-1/0:36:07

2009-09-22T21:31:14Z

Lzhang: video annotation

Invariance of tricolourability under R3 can be proved similarly (as for R2). There are no losses of colours locally (in some sense simpler). However, there are many more colour combinations, well just 5 up to permutation of colours, to check (perhaps this is the reason why Dror thinks the proof is boring).

Notes for AKT-090910-1/0:33:25

2009-09-22T21:25:59Z

Lzhang: video annotation

Cont'd: Invariance of tricolourability under R2. In particular, the subtlety (as viewed locally) about the total number of colours used (globally) is discussed.

Problem/Concern:
Sometimes, in a local picture, only 2 colours appear on one side of an isotopy move (e.g. R2) whereas all 3 colours appear on the other. One might worry that this could lead to the violation of the global rule.

Solution:
One can prove that a knot (consisting of only 1 connected piece of material in <math>\mathbb{R}^3</math>), which is coloured obeying the local rule of tricolourability, has at least 2 colours <math>\Leftrightarrow</math> it has all 3 colours. The proof relies on the fact that the same piece of material can change colour (from one colour to a 2nd colour) only by going 'under' a crossing, and whenever a crossing involves 2 colours it must involve a 3rd. (This argument fails, however, for links. Just consider two knots, one red and one blue say, placed side by side.)

Notes for AKT-090910-1/0:29:24

2009-09-22T21:25:53Z

Lzhang:

Invariance of tricolourability under R2. For each of the 2 directions (as in the proof for R1), there are cases (corresponding to different colourings of, say the top, 'external endpoints' of the local diagram modulo colour permutations; the colouring of all arcs will then be forced by the local rule of tricolourability and the fact that we are only dealing with three colours, and the cases which violate the global rule should be discarded) to check.

Terminology (for the purpose of this annotation) 
Local rule: at each crossing, of the 3 arcs involved, either 1 or all 3 colours appear. 
Global rule: all 3 colours must appear. 
(Together with the provision of 3 colours, knots which can be coloured obeying these rules, are called tricolourable as defined at 0:10:26 of this hour.)

Notes for AKT-090910-1/0:29:24

2009-09-22T20:43:23Z

Lzhang: video annotation

Notes for AKT-090910-1/0:26:16

2009-09-22T20:39:46Z

Lzhang:

Invariance of tricolourability under R1. There are 2 directions to check, i.e. (left) 3-colourable <math>\Rightarrow</math> (right) 3-colourable, and (left) colourable <math>\Leftarrow</math> (right) colourable. Side benefit: explanation of 'local' diagram notation.

Notes for AKT-090910-1/0:26:16

2009-09-22T20:13:52Z

Lzhang: video annotation

Invariance of tricolourability under R1. Side benefit: explanation of 'local' diagram notation.

Notes for AKT-090910-1/0:26:06

2009-09-22T20:13:44Z

Lzhang: video annotation

Sketch of proof that tricolourability is invariant under all 3 Reidemeister moves:

Notes for AKT-090910-1/0:23:54

2009-09-22T20:05:45Z

Lzhang: video annotation

Reducing the problem: via the Reidemeister Theorem, to proving that tricolourability is invariant under each of the 3 Reidemeister moves.

Notes for AKT-090910-1/0:18:44

2009-09-22T20:00:00Z

Lzhang: video annotation

Want to prove: tricolourability is a knot invariant. Quoting and explaining the Reidemeister Theorem, which is relied upon in the proof.

Notes for AKT-090910-1/0:16:25

2009-09-22T19:50:36Z

Lzhang: making the statement more precise

Proving: the (particular planar projection of the) trefoil knot is tricolourable whereas (the planar diagram of) the unknot is not.

Notes for AKT-090910-1/0:16:25

2009-09-22T19:46:17Z

Lzhang: video annotation

Proving: the trefoil knot is tricolourable whereas the unknot is not.

Notes for AKT-090910-1/0:10:26

2009-09-22T19:42:00Z

Lzhang: video annotation

Construction/definition of such an invariant (simplest for our purpose), tricolourability, on knot diagrams.

Notes for AKT-090910-1/0:08:18

2009-09-22T19:36:26Z

Lzhang: video annotation

Define invariant of knots. Our approach to prove Trefoil knot <math>\ne</math> Unknot: to construct an invariant that distinguishes these two knots.